2014 Mathematics SOL Institutes Grade Band: Geometry 1

Making Connections and Using Representations The purpose of the 2014 Mathematics SOL Institutes is to provide professional development focused on instruction that supports process goals for students in mathematics. Emphasis will be on fostering students’ ability to make mathematical connections and use effective and appropriate representations in mathematics. 2

Agenda I.Defining Representations and Connections II.Doing the Mathematical Task III.Looking at Student Work IV.Planning Mathematics Instruction V.Closing 3

Welcome! Draw a geometric shape that represents you. Be able to justify your choice. Group yourselves with others who have like attributes. Discuss your rationale for your shape and your group classification. 4

I. Defining Representation and Connection 5 1.Describe in words what the fourth arrangement would look like.

Defining Representation 6 2.Using the cubes, build the fourth arrangement.

Defining Representation 7 3.How many cubes are in the fourth arrangement? 4.How many cubes are in the fifth arrangement?

Defining Representation 8 5.How many cubes are in the 50 th arrangement? 6.How many cubes are in the n th arrangement?

What does it mean for students to use mathematical representations? Think/Pair/Share 9 Defining Representation

Mathematical Representations Students will represent and describe mathematical ideas, generalizations, and relationships with a variety of methods. Students will understand that representations of mathematical ideas are an essential part of learning, doing, and communicating mathematics. Students should move easily among different representations graphical, numerical, algebraic, verbal, and physical and recognize that representation is both a process and a product. 10 Virginia Department of Education. (2009). Introduction Mathematics Standards of Learning for Virginia Public Schools

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12 “Representations are useful in all areas of mathematics because they help us develop, share, and preserve our mathematical thoughts. They help to portray, clarify, or extend a mathematical idea by focusing on its essential features.” National Council of Teachers of Mathematics. (2000). Principles and Standards for School Mathematics. (p. 206). Reston, VA.

What does it mean for students to make mathematical connections? Think/Pair/Share 13 Defining Connection

Mathematical Connections Students will relate concepts and procedures from different topics in mathematics to one another and see mathematics as an integrated field of study. Through the application of content and process skills, students will make connections between different areas of mathematics and between mathematics and other disciplines, especially science. Science and mathematics teachers and curriculum writers are encouraged to develop mathematics and science curricula that reinforce each other. 14 Virginia Department of Education. (2009). Introduction Mathematics Standards of Learning for Virginia Public Schools

15 “Connections are useful because they help students see mathematics as a unified body of knowledge rather than a set of complex and disjoint concepts, procedures and processes. Real world contexts provide opportunities for students to connect what they are learning to their own environment. Their mathematics may also be connected to other disciplines which provides opportunities to enrich their learning.” National Council of Teachers of Mathematics. 2000, p Principles and Standards for School Mathematics. Reston, VA

Planning for the Use of Representations How do we plan for the purposeful use of representations and connections in the classroom? What questions do we need to consider? Brainstorm as many questions as you can, and share on your chart paper. 16

II. Doing the Mathematical Task Find the coordinates of the bus stop that is equidistant to three houses A, B, and C. Explain and justify. Find other houses that are equidistant to the bus stop. Explain and justify. 17 A B C x y

Doing the Mathematical Task Solve each part of the problem in two ways. Be prepared to share your solutions. 18

Orchestrating Mathematics Discussion 1.Anticipating 2.Monitoring 3.Selecting 4.Sequencing 5.Connecting 19

Doing the Mathematical Task In your groups, discuss: What mathematical representations could be used to solve this task? What mathematical connections can be made with this task? 20

III. Looking at Student Work For each student sample, identify: –The representation(s) used; –The connections that are evident; and –Any misconceptions evident in the student work. 21

Student Sample I Part a) Part b) 22

Questions to Consider Revisit your list of questions to consider when planning mathematics instruction. Is there anything we need to add? 23

Representation should be an important element of lesson planning. Teachers must ask themselves, “What models or materials (representations) will help convey the mathematical focus of today’s lesson?” - Skip Fennell 24 Fennell, F (Skip). (2006). Representation—Show Me the Math! NCTM News Bulletin. September. Reston, VA: NCTM

IV. Planning Mathematics Instruction Afternoon goal – Answer this question: How can we best integrate constructions into our Geometry curriculum? 25

Perpendicular Bisectors/Circumscribed Circles On the handout, complete the constructions for: –Perpendicular bisector of a segment –Circumscribed circle of a triangle While completing the constructions, think carefully about why these constructions work. 26

Perpendicular Bisectors/Circumscribed Circles At your table, discuss: How can we justify these constructions? When should we teach these constructions? When should we use the bus stop task? 27

Creating a construction Paper/Pencil with Compass and Straightedge Paper Folding Virtually with TestNav available on VDOE WebsiteVDOE Website Dynamic Geometry Software 28

Using Dynamic Geometry Software In tables, number from 1 – 4. Rearrange to be with like numbers. Watch for: 1: Teacher moves for connections 2: Student moves for connections 3: Teacher moves for representations 4: Student moves for representations 29

Using Dynamic Geometry Software Watch video In numbered groups, share your observations. Return to original tables. Share the consensus from the breakout groups. The Teaching Channel. Using Dynamic Geometry Software 30

Integrating Constructions Create a poster for your assigned construction. Include: –Actual construction –Mathematical connections –Justifications –When we should teach this construction 31

Gallery Walk Designate one person in your group as the expert to stay at your poster and explain. Remainder of the group participate in the gallery walk. 32

33 "Students representational competence can be developed through instruction. Marshall, Superfine, and Canty (2010, p. 40) suggest three specific strategies: 1. Encourage purposeful selection of representations. 2. Engage in dialogue about explicit connections among representations. 3. Alternate the direction of the connections made among representations." National Council for Teachers of Mathematics. (2014). Principles to Actions. (p. 26). Reston, VA

The Role of the Teacher Create a learning environment that encourages and supports the use of multiple representations Model the use of a variety of representations Orchestrate discussions where students share their representations and thinking Support students in making connections among multiple representations, to other math content and to real world contexts Van de Walle, J.A., Karp, K.S., Lovin, L.H. & Bay-Williams, J.M. (2013). Teaching Student-Centered Mathematics: Developmentally Appropriate Instruction for Grades 3-5 (2 nd ed.). (Vol. II). Pearson. 34

Role of the Student Create and use representations to organize, record, and communicate mathematical ideas Select, apply, and translate among mathematical representations to solve problems Use representations to model and interpret physical, social, and mathematical phenomena Van de Walle, J.A., Karp, K.S., Lovin, L.H. & Bay-Williams, J.M. (2013). Teaching Student-Centered Mathematics: Developmentally Appropriate Instruction for Grades 3-5 (2 nd ed.). (Vol. II). Pearson. 35

36 Students must be actively engaged in developing, interpreting, and critiquing a variety of representations. This type of work will lead to better understanding and effective, appropriate use of representation as a mathematical tool. National Council of Teachers of Mathematics. (2000) Principles and Standards for School Mathematics. (p. 206). Reston, VA.

V. Closing Revisit questions to consider for planning mathematical instruction. Which questions should we add? How does your list compare with the list on the handout? 37

Closing What practical implications does our work with constructions have for your classroom, school, or division? How will it inform instruction and pacing? Complete the current and ideal practice form. 38

Closing Share your current and ideal practice form at your table. 39